Make plane fold in curve (arc) while conserving its length
When you say length should be "conserved," do you mean this in an approximate sense or a mathematically strict sense?  Doing it in a strict sense might be more difficult.

I am a little confused when you are talking about using a bevel modifier but you're showing the use of an armature.  You could certainly use them together, but why not simply parent the plane to the bones of your armature with weight paints to control the influence of each bone in a graduated/curved fashion?

 

For this example I used linear gradient weight paints for each bone:





I am sure you could tweak the gradients to get the curvature more to your liking.

But maybe this isn't what you had in mind?